Bernoulli, Daniel
,
Hydrodynamica, sive De viribus et motibus fluidorum commentarii
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SECTIO DECIMA.
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tate ſuo pondere aërem infimum expellet per foramen, qua aër in vaſe pro-
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poſito ſua elaſticitate ſe ipſum expellit. </
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<
s
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echoid-s6610
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xml:space
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">In priori autem caſu ejicitur veloci-
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tate quæ debetur ipſi altitudini cylindri, ergo & </
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<
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">in poſteriori. </
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<
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xml:space
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">Notandum
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autem eſt, altitudinem quam pro cylindro finximus, perpetuo eandem eſſe,
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quia aëris elaſticitas & </
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<
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echoid-s6613
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">denſitas in eadem ratione diminuuntur, calorem autem
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non mutari ponimus. </
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<
s
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echoid-s6614
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xml:space
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">Igitur ſi altitudo aëris homogenei (quæ à calore aëris
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interni pendet) dicatur A, effluet aër conſtanter velocitate √ A. </
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<
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">Nec tamen,
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quod calculus oſtendit, vas ipſum unquam evacuatur, quia aër effluens fit
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continue rarior, quod ut æquatione comprehendamus, ponemus denſitatem ſeu
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quantitatem aëris à fluxus initio = 1; </
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<
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">denſitatem ſeu quantitatem aëris poſt de-
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finitum tempus reſidui = x, tempusque ipſum = t, erit, quia velocitas
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conſtans eſt, - d x = a x d t, ubi per a intelligitur quantitas conſtans defi-
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nienda ex magnitudine vaſis, amplitudine foraminis & </
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<
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">altitudine A: </
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">hinc
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{- dx/x} = adt & </
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<
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xml:space
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<
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xml:space
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">reperitur autem valor coëfficientis a hoc modo.
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</
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<
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">Quia poſitum à nobis fuit - d x = a x d t; </
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<
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xml:space
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">erit ab initio effluxus - dx = a d t. </
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Jam mutetur elementum primum (- d x) in cylindrum foramini ceu baſi ſu-
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perinſtructum; </
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<
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xml:space
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">erit autem altitudo iſtius cylindruli = - L d x, ſi L ſit altitu-
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do cylindri ſuper eodem foramine extructi & </
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<
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ſito capacitatem habentis: </
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<
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">hæc porro longitudo - L d x illa eſt, quæ tem-
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puſculo d t percurritur, & </
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">quia poni ſolet tempuſculum æquale ſpatio percur-
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ſo diviſo per velocitatem, erit hic d t = {- L d x/√ A}; </
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<
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xml:space
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æquatione - d x = a d t & </
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">habebitur - d x = {- a L d x/√A}, ſive a = {√A/L}. </
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proinde æquatio finalis hæc: </
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log. </
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<
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xml:space
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">{1/x} = {t√A/L}.</
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<
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">Si tempus exprimere lubeat per certum minutorum ſecundorum nu-
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merum, quem vocabimus n, & </
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<
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">intelligatur per s ſpatium quod mobile ab-
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ſolvit cadendo libere à quiete intra unum minutum ſecundum, erit ponen-
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dum t = 2n√s, ſicque fiet
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log. </
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<
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xml:space
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">{1/x} = {2n√As/L}.</
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